Pre Calc: 4sin^3 (x) - 4si2^3 (x) - 3sin(x) +3 = 0

[Math wiz ya rite 09]

Could someone show me step by step how to solve this so I can apply it to the rest of my homework problems. Thank you!

Solve for x:

4sin^3 (x) - 4si2^3 (x) - 3sin(x) +3 = 0

for x in the interval [(-3pi)/2, pi]

 

[hypoid]

what's a 4si2^3(x)

 

[soroban]

Re: Student in Pre Calc in need of assistence

Hello, Math wiz ya rite 09!

Solve for \(\displaystyle x:\;\;4\sin^3x\,-\,4\sin^2x\,-\,3\sin x\,+\,3\;=\;0\;\) for \(\displaystyle x\,\in\,\left[-\frac{3\pi}{2},\:\pi\right]\)

Factor: \(\displaystyle \:4\sin^2x(\sin x\,-\,1)\,-\,3(\sin x\,-\,1)\;=\;0\)

Factor: \(\displaystyle \\sin x\,-\,1)(4\sin^2x\,-\,3)\;=\;0\)


And solve the two equations:

\(\displaystyle (1)\;\sin x\,-\,1\:=\:0\;\;\Rightarrow\;\;\sin x\:=\:1\)

. . Therefore: \(\displaystyle \L\:\fbox{x\:=\:-\frac{3\pi}{2},\:\frac{\pi}{2}}\)


\(\displaystyle (2)\;4\sin^2x\,-\,3\:=\:0\;\;\Rightarrow\;\;\sin^2x \,=\,\frac{3}{4}\;\;\Rightarrow\;\;\sin x \,=\,\pm\frac{\sqrt{3}}{2}\)

. . Therefore: \(\displaystyle \L\:\fbox{x\:=\:-\frac{4\pi}{3},\:-\frac{2\pi}{3},\:-\frac{\pi}{3},\:\frac{\pi}{3},\:\frac{2\pi}{3}}\)